Non-linearity in Davenport-Schinzel Sequences Seth Pettie - - PowerPoint PPT Presentation

Non-linearity in Davenport-Schinzel Sequences

Seth Pettie

University of Michigan

Seth Pettie

Isomorphism and Subsequences

Political Isomorphism

BUSH BUSH is isomorphic to GO GORE C is isomorphic to A THOMAS THOMAS is isomorphic to SOUTER CI CIA,N ,NSA,DOD is not isomorphic to NSF,EPA,NIH IH

Happiness via Subsequences

WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P

Seth Pettie

Isomorphism and Subsequences

Political Isomorphism

BUSH BUSH is isomorphic to GO GORE C is isomorphic to A THOMAS THOMAS is isomorphic to SOUTER CI CIA,N ,NSA,DOD is not isomorphic to NSF,EPA,NIH IH

Happiness via Subsequences

WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P

Seth Pettie

Isomorphism and Subsequences

Political Isomorphism

BUSH BUSH is isomorphic to GO GORE C is isomorphic to A THOMAS THOMAS is isomorphic to SOUTER CI CIA,N ,NSA,DOD is not isomorphic to NSF,EPA,NIH IH

Happiness via Subsequences

WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P

Seth Pettie

Isomorphism and Subsequences

Political Isomorphism

BUSH BUSH is isomorphic to GO GORE C is isomorphic to A THOMAS THOMAS is isomorphic to SOUTER CI CIA,N ,NSA,DOD is not isomorphic to NSF,EPA,NIH IH

Happiness via Subsequences

WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P

Seth Pettie

Isomorphism and Subsequences

Political Isomorphism

BUSH BUSH is isomorphic to GO GORE C is isomorphic to A THOMAS THOMAS is isomorphic to SOUTER CI CIA,N ,NSA,DOD is not isomorphic to NSF,EPA,NIH IH

Happiness via Subsequences

WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH TH_WHO I E _ V A BE R TA TARJAN W W O R K S HO P

Seth Pettie

Isomorphism and Subsequences

Political Isomorphism

BUSH BUSH is isomorphic to GO GORE C is isomorphic to A THOMAS THOMAS is isomorphic to SOUTER CI CIA,N ,NSA,DOD is not isomorphic to NSF,EPA,NIH IH

Happiness via Subsequences

WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH_WHO O LD_ R_ VE ? TARJAN FOR PR EZ ?

Seth Pettie

Isomorphism and Subsequences

Political Isomorphism

BUSH BUSH is isomorphic to GO GORE C is isomorphic to A THOMAS THOMAS is isomorphic to SOUTER CI CIA,N ,NSA,DOD is not isomorphic to NSF,EPA,NIH IH

Happiness via Subsequences

WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? WITH_WHOM_WOULD_I_RATHER_HAVE_A_BEER? TH_WHO O LD_ R_ VE ? T TARJAN N FO FOR PR EZ ?

Seth Pettie

Definitions

x⊂y : x is isomorphic to a subsequence of y Ex(σ,n) = max |S| : S ∈ {1,…,n}* σ ⊄ S

S is |σ|-regular (technical condition)

How fast does Ex(σ,n) grow as a function of n?

Seth Pettie

Original application: lower envelopes

(1) Give each object (line segment, quadratic, etc.) a symbol (2) Map the lower envelope to a sequence |S| (3) Show |S| ≤ Ex(σ,n) for some forbidden subseq. σ this sequence does not contain ababa

Seth Pettie

Original motivation: lower envelopes

(1) Give each object (line segment, quadratic, etc.) a symbol (2) Map the lower envelope to a sequence |S| (3) Show |S| ≤ Ex(σ,n) for some forbidden subseq. σ standard case: σ = ababab…a “order k Davenport-Schinzel sequence” length k+2

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Splay trees and Davenport-Schinzel sequences

Amortized analysis: Normally pay for time consuming ops with a reduction in potential

Seth Pettie

Splay trees and Davenport-Schinzel sequences

New kind of amortized analysis: Label nodes that cannot be paid for by other means Transcribe the labels as a sequence S: |S| ≤ Ex(σ,n) In [SODA’08] σ = abaabba or abababa

• Thm. n deque operations take O(nα*(n)) time

Seth Pettie

Splay trees and Davenport-Schinzel sequences

New kind of amortized analysis: Label nodes that cannot be paid for by other means Transcribe the labels as a sequence S: |S| ≤ Ex(σ,n)

…

A much better way to end the proof:

… where Ex(σ,n) = O(n)

Seth Pettie

Standard Davenport-Schinzel seqs.

α = α(n) α is the inverse-Ackermann function

2n-1 Ex(abab, n) n Ex(aba, n)

trivial

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Standard Davenport-Schinzel seqs.

α = α(n) α is the inverse-Ackermann function

Θ(nα) Ex(ababa, n) 2n-1 Ex(abab, n) n Ex(aba, n)

trivial Hart-Sharir

Seth Pettie

Standard Davenport-Schinzel seqs.

α = α(n) α is the inverse-Ackermann function

Θ(n2α) Ex(ababab, n) Θ(nα) Ex(ababa, n) 2n-1 Ex(abab, n) n Ex(aba, n)

trivial Hart-Sharir

Agarwal-Sharir-Shor

Seth Pettie

Standard Davenport-Schinzel seqs.

α = α(n) α is the inverse-Ackermann function

n exp(Θ(α3)) Ex(ababababab, n) n exp(O(α2log α)) Ex(ababababa, n) n exp(Θ(α2)) Ex(abababab, n) n exp(O(αlog α)) Ex(abababa, n) Θ(n2α) Ex(ababab, n) Θ(nα) Ex(ababa, n) 2n-1 Ex(abab, n) n Ex(aba, n)

trivial Hart-Sharir

Agarwal-Sharir-Shor

Seth Pettie

Standard Davenport-Schinzel seqs.

α = α(n) α is the inverse-Ackermann function

n exp(O(α|σ|)) Ex(σ, n) n exp(Θ(α3)) Ex(ababababab, n) n exp(O(α2log α)) Ex(ababababa, n) n exp(Θ(α2)) Ex(abababab, n) n exp(O(αlog α)) Ex(abababa, n) Θ(n2α) Ex(ababab, n) Θ(nα) Ex(ababa, n) 2n-1 Ex(abab, n) n Ex(aba, n)

trivial Hart-Sharir

Agarwal-Sharir-Shor

Klazar

Seth Pettie

Two-Letter Forbidden Subsequences

[Adamec-Klazar-Valtr] Ex(abbaab,n) = O(n)

The Two-Letter Theorem: For any σ ∈ {a,b}* Ex(σ,n) = ω(n) if and only if ababa ⊂ σ

(i.e., there is only one “cause” of superlinearity over two symbols)

Seth Pettie

The Three-Letter Theorem

[Klazar-Valtr] For σ ∈ {a,b,c}* Ex(σ,n) = O(n)

unless…

ababa ⊂ σ or abcacbc ⊂ σ or abcbcac ⊂ σ

• r their reversals

non-linear status still open

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uaaw,n) = O(n) and Ex(v,n) = O(n) Ex(uavaw,n) = O(n) (3) If Ex(uaawa) = O(n) Ex(uabiawabi) = O(n) For Example: Ex(aabbaabcdddcefgfefgcccbbccdd) = O(n)

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uaawa) = O(n) Ex(uabiawabi) = O(n) For Example: Ex(aabbaabcdddcefgfefgcccbbccdd) = O(n)

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n)

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aaaa

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aabbaabbb

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aabbaabcccccbbcc

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aabbaabcdddccccbbccdd

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aabbaabcdddccccbbccdd ee

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aabbaabcdddccccbbccdd effef

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aabbaabcdddccccbbccdd efgfefg

uw and v have disjoint alphabets

Seth Pettie

Recipe for linear forbidden sequences

[Klazar-Valtr] (1) Ex(ai,n) = O(n) (2) If Ex(uw,n) = O(n) and Ex(v,n) = O(n) Ex(uvw,n) = O(n) (3) If Ex(uawa,n) = O(n) Ex(uabiwabi) = O(n) aabbaabcdddcefgfefgcccbbccdd efgfefg

uw and v have disjoint alphabets

Seth Pettie

More than one cause of non-linearity

[Klazar]

σ is a sequence without repetitions (x,y) is in G(σ) iff xyyx ⊂ σ or yxyx ⊂ σ If G(σ) is strongly connected then

Ex(σ,n) = Ω(nα(n))

Seth Pettie

More than one cause of non-linearity

[Klazar]

σ is a sequence without repetitions (x,y) is in G(σ) iff xyyx ⊂ σ or yxyx ⊂ σ If G(σ) is strongly connected then

Ex(σ,n) = Ω(nα(n))

G(ababa) G(abcbadadbcd)

• nly two

examples known

Seth Pettie

Another cause of non-linearity

[Klazar]

σ is a sequence without repetitions (x,y) is in G’(σ) iff xyyx ⊂ σ or yxyx ⊂ σ If G’(σ) is strongly connected then

Ex(σ,n) = Ω(nα(n)) Ω(n2α(n))

G’(ababab) G’(abcbadadbecfcfedef)

• nly two

examples known

Seth Pettie

• Defn. Φ = minimal non-linear forbidden seqs.

What we know about Φ:

ababa ∈ Φ |Φ| ≥ 2 (the other a subseq of abcbadadbcd)

Seth Pettie

• Defn. Φ = minimal non-linear forbidden seqs.

What we know about Φ:

ababa ∈ Φ |Φ| ≥ 2 (the other a subseq of abcbadadbcd)

Q: Is |Φ| infinite? A: Still Open. But we have a candidate!

Seth Pettie

• Defn. Φ = minimal non-linear forbidden seqs.

What we know about Φ:

ababa ∈ Φ |Φ| ≥ 2 (the other a subseq of abcbadadbcd)

Q: Is |Φ| infinite? A: Still Open. But we have a candidate! Q: How big is it Φ? A: New result: |Φ| ≥ 5

Seth Pettie

Constructing Sequences

T(1,j) : a binary tree with height j+1 j distinct letters at each leaf

Seth Pettie

Constructing Sequences

T(1,j) : a bin. tree w/height j+1, j letters at each leaf

ith letter at a leaf added to label of ith ancestor

Seth Pettie

Constructing Sequences

T(1,j) : a bin. tree w/height j+1, j letters at each leaf

ith letter at a leaf added to label of ith ancestor

Seth Pettie

Constructing Sequences

T(1,j) : a bin. tree w/height j+1, j letters at each leaf

ith letter at a leaf added to label of ith ancestor

Seth Pettie

Constructing Sequences

T(1,j) : a bin. tree w/height j+1, j letters at each leaf

ith letter at a leaf added to label of ith ancestor

Seth Pettie

Constructing Sequences

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T(k,j) : composition of j T(k-1, ⋅) trees, j distinct letters at each leaf.

Seth Pettie

the ith letter at a leaf is assigned to the ith (k-1)-node ancestor of the leaf

T(k,j) : composition of j T(k-1, ⋅) trees, j distinct letters at each leaf.

Seth Pettie

the ith letter at a leaf is assigned to the ith (k-1)-node ancestor of the leaf …and the T(k-1, ⋅) trees are defined in terms of their leaf labels…

T(k,j) : composition of j T(k-1, ⋅) trees, j distinct letters at each leaf.

Seth Pettie

Constructing Sequences

v1,v2,…,vn : nodes listed in postorder L(v) : the label of v in reverse order The final sequence: Σ = L(v1),L(v2),…,L(vn)

The sequence for T(1,4) : cba fed da ihg lkj jg kheb

• nm rqp pm uts xwv vs wtqn

xurolifx …

Seth Pettie

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Forbidden subseq: ababa

Σ is (ababa)-free:

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Forbidden subseq: ababa

Σ is (ababa)-free:

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Forbidden subseq: ababa

Σ is (ababa)-free:

these are in the wrong order!

Seth Pettie

Forbidden subseq: abcaccbc

Σ is (abcaccbc)-free:

necessarily a common ancestor

Seth Pettie

Forbidden subseq: abcaccbc

Σ is (abcaccbc)-free:

necessarily different nodes

Seth Pettie

Forbidden subseq: abcaccbc

Σ is (abcaccbc)-free:

necessarily different nodes

Seth Pettie

Forbidden subseq: abcaccbc

“a” does not appear in the final contradiction

(an implied occurrence of bcbcbc)

Why is it necessary?

necessarily different nodes

the “binder”

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Forbidden subseq: abcdeaebdce

Σ is (abcdeaebdce)-free:

necessarily different nodes

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Forbidden subseq: abcdeaebdce

Σ is (abcdeaebdce)-free:

necessarily different nodes

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Forbidden subseq: abcdeaebdce

Σ is (abcdeaebdce)-free:

necessarily different nodes

“guard” “binder” “trapper” “trapped elements”

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Forbidden subseq: abcdeaebdce

Succinct Encoding: ♥♦♠♠♣

♥ : a = binder ♦ : b = guard ♠ : c = 1st trapped ♠ : d = 2nd trapped ♣ : e = trapper

necessarily different nodes

“guard” “binder” “trapper” “trapped elements”

Seth Pettie

All of these encodings make sense & work: ♥♦♠♠♣ ♦♥♠♠♣ ♥♠♦♠♣ These don’t: ♦♠♠♥♣ ♥♠♠♦♣ ♥♣♦♠♣

the binder doesn’t bind (but this can be fixed!) the guard doesn’t guard this doesn’t make any sense

Seth Pettie

Forbidden subseq: abcdeafefbdcf

Encoding: ♦♠♠♥♣

 : a = half-binder ♦ : b = guard ♠ : c = 1st trapped ♠ : d = 2nd trapped ♥ : e = binder ♣ : f = trapper “guard” “binder” “trapped elements” “trapper” “half-binder”

Seth Pettie

Forbidden subseq: abcdeafegfhgihjijbdcj

Half-binders can be “daisy-chained”

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Seventeen legal encodings

♥♠(♦♠♣) ♠♥(♦♠)♣ ♥♠(♦♠)♣ ♣♠♥♣ ♥♣♠♣ ♦♠♠♥♣ ♠♥(♦♠♣) ♦♠♥♠♣ ♥♦♠♠♣ ♦♠♠♥♣ ♥♠♦♠♣ ♦♠♥♠♣ ♦♥♠♠♣ ♠♦♠♥♣ ♠(♦♠)♥ ♠♦♥♠♣ ♠♥♦♠♣

Seth Pettie

Some open problems

Are there infinitely many “causes” of non-linearity? Are there any more linear seqs. to be discovered? For each c, is there an (ababa)-free σ such that: Ex(σ,n) = n exp(αc(n))